I Was wondering about how you find the residue for an essential singularity. I was under the impression that finding the first non zero term in the principle series of the Laurent series only works for isolated singularities?
2026-03-24 20:41:27.1774384887
Finding the residue of essential singularities
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No. By definition, the residue is the coefficient of $\frac1z$ of the Laurent series. For instance, if you have to compute the residue at $0$ of $\sin\left(\frac1z\right)$ you not that$$\sin\left(\frac1z\right)=\frac1z-\frac1{3!z^3}+\cdots$$and so the residue is $1$ (the coefficient of $\frac1z$).